Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.220544, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]
[Out]
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Rubi in Sympy [A] time = 6.40103, size = 34, normalized size = 0.85 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{c x^{2} + d \sqrt{a + b x^{4}}}} \right )}}{a \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.16237, size = 50, normalized size = 1.25 \[ \frac{\sqrt{-\frac{1}{c}} \cot ^{-1}\left (\frac{\sqrt{-\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}+c x^2}}{x}\right )}{a} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]
[Out]
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Maple [F] time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}{\frac{1}{\sqrt{c{x}^{2}+d\sqrt{b{x}^{4}+a}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{4}\right ) \sqrt{c x^{2} + d \sqrt{a + b x^{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="giac")
[Out]